The concepts of ``circuits'' and ``ports'' from classical
circuit/network theory [35] are very useful for
partitioning complex systems into self-contained sections
having well-defined (small) interfaces. For example, it is typical in
analog electric circuit design to drive a high-input-impedance stage
from a low-output-impedance stage (a so-called ``voltage transfer''
connection). This large impedance ratio allows us to neglect
``loading effects'' so that the circuit sections (stages) can be
analyzed separately.

As discussed in Chapter 7 (§7.2), the equivalent
circuit for a force-driven mass is shown in Fig.1.8. The
mass
is represented by an inductor
. The driving
force
is supplied via a voltage source, and the mass
velocity
is the loop current.

Figure:
Electrical equivalent circuit of the
force-driven mass in Fig.1.6.

As also discussed in Chapter 7 (§7.2), if two physical
elements are connected in such a way that they share a common
velocity, then they are said to be formally connected in
series. The ``series'' nature of the connection becomes more clear
when the equivalent circuit is considered.

Figure 1.9:
A mass and spring connected in series.

For example, Fig.1.9 shows a mass connected to one
end of a spring, with the other end of the spring attached to a rigid
wall. The driving force
is applied to the mass
on the left so that a positive force results in a positive mass
displacement
and positive spring displacement (compression)
. Since the mass and spring displacements are physically the
same, we can define
. Their velocities are
similarly equal so that
. The equivalent circuit
has their electrical analogs connected in series, as shown in
Fig.1.10. The common mass and spring velocity
appear as a single current running through the inductor (mass) and
capacitor (spring).

Figure:
Electrical equivalent circuit of the
series mass-spring driven by an external force diagrammed in
Fig.1.9.

By Kirchoff's loop law for circuit analysis, the sum of all voltages
around a loop equals zero.2.13 Thus, following
the direction for current
in Fig.1.10, we have
(where a minus sign is used when the
current enters the `
' sign of the element--this means the forces
in Fig.1.10 are positive when pointing to
the right),
or

Thus, the equivalent circuit agrees with our direct physical analysis
that the applied force
is equal at all times to
the sum of the mass inertial force
and spring
force
, i.e.,